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Cheat Sheet Final Exam Di erentiation rules f g cid 48 f cid 48 g cid 48 f g cid 48 f cid 48 g cid 48 cf cid 48 cf cid 48 f g cid 48 f cid 48 g f g cid 48 cid 48 cid 48 g cid 48 f f g f g g2 f g x cid 48 f cid 48 g x g cid 48 x The most common derivatives c cid 48 0 xn cid 48 nxn 1 ex cid 48 ex ax cid 48 ax ln a ln x cid 48 1 x loga x cid 48 1 x ln a sin x cid 48 cos x cos x cid 48 sin x tan x cid 48 sec2 x sec x cid 48 sec x tan x csc x cid 48 csc x cot x cot x cid 48 csc2 x arcsin x cid 48 1 1 x2 arccos x cid 48 1 1 x2 arctan x cid 48 1 1 x2 cot 1 x cid 48 1 1 x2 sec 1 x cid 48 1 x x2 1 csc 1 x cid 48 1 x x2 1 Two important limits limx 0 sin x x 1 limx 0 1 cos x x 0 Implicit Di erentiation When the relation between the dependent variable y and the independent variable x is given by an equation we can nd y cid 48 by taking derivatives of both In other words y sides of the equation with respect to x treating y as a function of x should be treated as the variable u in the u substitution scheme Related Rates Sometimes we wish to nd the rate of change in time of a certain quan tity say A but we have no way of doing it directly Usually we can nd the rate of a change of a related quantity B In order to nd the rate of change of A it is necessary rst to nd an equation relating A and B and then proceed to di erentiate said equation with re spect to time treating both A and B as functions of time like in the u substitution scheme Logarithmic Di erentiation In order to nd the derivative of a function of the form y f x g x take natural logarithms in both sides of the previous equation and then pro ceed to di erentiate using implicit di erentiation Rolle s Theorem If f is a continuous function in a b di erentiable in a b such that f a f b then there exists a number c in a b such that f cid 48 c 0 Mean Value Theorem If f is a continuous function in a b and di erentiable in a b then there exists a number c in a b such that f cid 48 c f b f a b a 1 Relations between a function and its derivatives A function f is increasing at each point x where f cid 48 x 0 On the other hand it is decreasing at each point x where f cid 48 x 0 It is concave up at x if f cid 48 cid 48 x 0 and concave down at x if f cid 48 cid 48 x 0 Important De nitions Let f be a function and c a point of its domain D 1 We say that c is an absolute maximum of f if f x f c for all x in D On the other hand we say that c is an absolute minimum of f if f x f c for all x in D 2 We say that c is a local maximum of f if f x f c for the points x near c and that c is a local minimum of f if f x f c for the points x near c 3 We say that c is a critical point of f if f cid 48 c 0 or if f cid 48 c doesn t exist 4 We say that c is an in ection point of f if f changes concavity at c The Closed Interval Method Given a continuous function f in a closed interval a b we can nd its absolute maximum and its absolute minimum value by following these steps 1 Find the critical points of f in a b 2 Evaluate f at the critical points 3 Evaluate f at the endpoints of the interval 4 Compare values and decide First Derivative Test Let f be a function and let c be a critical point of f If f cid 48 changes sign from negative to positive at c then c is a local minimum of f On the other hand if f cid 48 changes sign from positive to negative at c then c is a local maximum of f If f cid 48 doesn t change signs at c then c isn t either a local maximum nor a local minimum of f Second Derivative Test Let f be a function and let c be a point such that f cid 48 c 0 If f cid 48 cid 48 c 0 then c is a local maximum of f If f cid 48 cid 48 c 0 then c is a local minimum of f De nitions Whenever we wish to compute a limit of the form and limx a f x 0 limx a g x 0 we say that our original limit is an indeterminate 0 On the other hand if limx a f x and limx a g x we refer form of type 0 to our original limit as an indeterminate form of type L Hospital s Rule If the limit lim x a f x g x lim x a f x g x is an indeterminate form of type 0 0 or of type then lim x a f x g x lim x a f cid 48 x g cid 48 x Slant Asymptotes We say that the line y mx b is a slant asymptote of the function f x if lim x f x mx b 0 This usually occurs in rational functions whenever the degree of the numerator is 1 higher than the degree of the denominator Guidelines for sketching a curve A Domain B Intercepts C Symmetry D Asymptotes Vertical Horizontal Slant E Intervals of increase and decrease F Local maxima and local minima G Concavity and points of in ection H Sketch the curve Good luck on the Final

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